Constructions of diagonal quartic and sextic surfaces with infinitely many rational points

TL;DR

This paper constructs new infinite families of diagonal quartic and sextic surfaces over the rationals that have infinitely many rational points, including some with minimal Picard number, expanding the understanding of rational solutions on higher-degree surfaces.

Contribution

It introduces explicit infinite families of diagonal quartic and sextic surfaces with infinitely many rational points, including those with Picard number one, which were previously unknown.

Findings

Constructed infinite families of diagonal quartic surfaces with rational points.

Presented sextic surfaces with infinitely many rational points.

Identified surfaces with Picard number one and infinite rational solutions.

Abstract

In this note we construct several infinite families of diagonal quartic surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where $a,b,c,d\in\Z\setminus\{0\}$ with infinitely many rational points and satisfying the condition $abcd\neq \square$. In particular, we present an infinite family of diagonal quartic surfaces defined over $\Q$ with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type $ax^6+by^6+cz^6+dw^i=0$, $i=2$, $3$, or $6$, with infinitely many rational points.